Example: Parametric Bootstrap estimate of the mean of a Normal distribution with known standard deviation

Imagine that we wish to estimate the true depth of a well using some sort of sonic probe. The probe has a known standard error s = 0.2 metres i.e. s is the standard deviation of the normally distributed variation of results the probe will produce when repetitively measuring the same depth. In order to estimate this depth we take n separate measurements. These measurements have a mean of  metres. The parametric Bootstrap model would take the average of n Normal(, s) distributions to estimate the true mean m of the distribution of possible measurement results, i.e. the true well depth. From Central Limit Theorem, we know that this calculation is equivalent to:

which is the standard classical statistics equation in this situation.


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